Phillips Brooks

John Phillips

Legal analysis is a multistep process that performs the complex tasks of identifying real world activities through the lens of a predefined legal code and sets of laws. Based on a brief understanding of an issue a legal professional must make a recommendation on where to apply limited researching resources. We employ Naïve Bayes models in a parallel, nonmutually exclusive alignment towards the ...

Brooks' Theorem and Beyond

We collect some of our favorite proofs of Brooks’ Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analo...

Books in Running Brooks

No Quantum Brooks' Theorem

First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed...

Sectoral Phillips curves and the aggregate Phillips curve

Sector-level Phillips curves are estimated in French data. There is considerable heterogeneity across sectors, with vastly different estimates of the backward looking component of inflation and the duration of nominal rigidities. A multi-sector model of inflation dynamics is calibrated on the basis of these sectoral estimates. Aggregate inflation, simulated on the basis of heterogeneous sectors...